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Problems of Information Transmission

, Volume 54, Issue 2, pp 101–115 | Cite as

On the Smallest Size of an Almost Complete Subset of a Conic in PG(2, q) and Extendability of Reed–Solomon Codes

  • D. BartoliEmail author
  • A. A. Davydov
  • S. Marcugini
  • F. Pambianco
Coding Theory
  • 21 Downloads

Abstract

Abstract—In the projective plane PG(2, q), a subset S of a conic C is said to be almost complete if it can be extended to a larger arc in PG(2, q) only by the points of C \ S and by the nucleus of C when q is even. We obtain new upper bounds on the smallest size t(q) of an almost complete subset of a conic, in particular,
$$t(q) < \sqrt {q(3lnq + lnlnq + ln3)} + \sqrt {\frac{q}{{3\ln q}}} + 4 \sim \sqrt {3q\ln q} ,t(q) < 1.835\sqrt {q\ln q.} $$
The new bounds are used to extend the set of pairs (N, q) for which it is proved that every normal rational curve in the projective space PG(N, q) is a complete (q+1)-arc, or equivalently, that no [q+1,N+1, q−N+1]q generalized doubly-extended Reed–Solomon code can be extended to a [q + 2,N + 1, qN + 2]q maximum distance separable code.

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Copyright information

© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  • D. Bartoli
    • 1
    Email author
  • A. A. Davydov
    • 2
  • S. Marcugini
    • 1
  • F. Pambianco
    • 1
  1. 1.Department of Mathematics and Computer SciencesUniversità degli Studi di PerugiaPerugiaItaly
  2. 2.Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

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